I found the account in John of Jesus washing the disciples feet to have some interesting features in the JST as compared to the KJV.
4 He riseth from supper, and laid aside his garments; and took a towel, and girded himself.
5 After that he poureth water into a bason, and began to wash the disciples’ feet, and to wipe them with the towel wherewith he was girded.
8 Peter saith unto him, Thou shalt never wash my feet. Jesus answered him, If I wash thee not, thou hast no part with me.
10 Jesus saith to him, He that is washed needeth not save to wash his feet, but is clean every whit: and ye are clean, but not all. (John 13:4-10)
8 Peter saith unto him, Thou needest not to wash my feet. Jesus answered him, If I wash thee not, thou hast no part with me.
9 Simon Peter saith unto him, Lord, not my feet only, but also my hands and my head.
10 Jesus saith to him, He that has washed his hands and his head, needeth not save to wash his feet, but is clean every whit; and ye are clean, but not all. Now this was the custom of the Jews under their law; wherefore, Jesus did this that the law might be fulfilled. (JST John 13:8-10)
In the KJV, Peter seems extra stubborn and uncharacteristically rebellious, saying,” Thou shalt never wash my feet.” His tone is more like what we would imagine Judas Iscariot having. But the JST substantially softens Peter’s words, showing us that he may have been more concerned about the utility of the act. Perhaps Peter’s feet had already been washed and a second washing seemed superfluous.
Also, in the KJV, Jesus’s words, “If I wash thee not, thou hast no part with me” seem petulant, but with Peter’s words moderated, Jesus’s words, though unchanged, seem more explanatory about how the washing brings them together in unity. If not washing Peter’s feet means Peter has no part with Christ, then washing Peter’s feet means Peter does have part with Christ. (Part with Christ in what manner?)
In the KJV, Jesus says “He that is washed needeth not save to wash his feet, but is clean every whit..” in the JST, however, Jesus says, “He that has washed his hands and his head, needeth not save to wash his feet, but is clean every whit..” This is interesting; first Peter thinks his feet are already washed and then Jesus insists they aren’t, and when Peter thinks his hands and head need washing, Jesus says they are already washed. This is a good indication that Jesus is speaking symbolically instead of literally. I think perhaps having head and hands clean symbolized pure thoughts and pure works, and washing one’s feet symbolized pure direction and goals, corresponding to the Lord's work to bring about the immortality and eternal life of man.
It must have been comforting to hear Jesus say, “ye are clean, but not all,” about Peter’s progress to that point. He was clean but not perfect. I suppose that when we are trying to live the gospel, keep the commandments, and repent of our sins, that would be Jesus’s message to us too. It would lift our sights higher and build desire in us to be further purified.
The JST also tells us that the washing of the feet was a requirement of Jewish law and that Jesus was fulfilling that law. This made me curious, and I looked up on the internet to see what the feet washing requirements were in the Jewish law, but I didn’t find anything that required it in conjunction with the Passover. However, I read in the Jewish Encyclopedia on washing that feet washing was part of more extensive cleansing requirements of the high priests to perform after sending off the scapegoat, and the same extensive cleansing (including feet washing) was required of the one who was to lead away the scapegoat. It is possible that Jesus was cleansing the disciples in the context of the Day of Atonement, in anticipation of being led away by one a disciple who would betray him to be crucified. Also, washing of hands and feet were to be done by those who wished to perform priestly duties, or washing of feet might be a simple gesture according to the law of hospitality to travelers. Whatever it means, it seems the Joseph Smith Translation restored knowledge of yet another way Jesus fulfilled the law during his mortal ministry, even if we don’t have knowledge of what exactly that law was. While we perhaps don’t need to know what that law was, our dispensation receives greater benefit from the instructions Jesus gave to cleanse the head, hands, and feet (thoughts, deeds, and goals) so that we can have part with Him.